Integrable Systems Seminar: Dzhamay -- Discrete Schlesinger Transformations and Difference Painlevé Equations

A/Prof Anton Dzhamay from the University of Northern Colorado will give a talk on
Wednesday 6th August at 1pm in Bosch Lecture Theatre 2.  

All welcome! 

TITLE: Discrete Schlesinger Transformations and Difference Painlevé Equations 

ABSTRACT: The main goal of the talk is to explain and compare the isomonodromic
approaches to the study of Painlevé equations in the differential and difference cases.
In the first part of the talk I will describe a discrete version of isomonodromic
deformations of Fuchsian systems, called Schlesinger transformations.  Similarly to the
continuous case, such transformations give a large number of commuting flows on the
moduli space of Fuchsian system (of a given spectral type), but in the discrete case we
can write explicit equations governing the dynamic; we call these equations discrete
Schlesinger equations.  These equations can, again in exact parallel with the continuous
case, be written in a Hamiltonian form on a slightly larger decomposition space (or the
eigenvector space) of Fuchsian systems.  When our phase space is two-dimensional,
discrete Schlesinger transformations reduce to difference analogues of Painlevé
equations.  In the second part of the talk I will consider one example of such reduction
that gives a difference Painlevé equation d-P(A_{2}^{(1)*}).  I will emphasize the role
played by Sakai’s geometric theory of discrete Painlevé equations in understanding the
structure of this reduction, and in comparing discrete Schlesinger dynamic with other
examples of difference Painlevé dynamic of the same type.